a sin13pi12 B sin7pi12 Use a sum or difference formula or a

a) sin(13pi/12)

B) sin(-7pi/12)

Use a sum or difference formula or a half angle formula

to determine the value of the trigonometric functions. Give exact answers. Do not use decimal numbers.

Solution

a) sin(13pi/12) = sin(10pi/12 + 3p/12)

Use sin(A+B) = sinA cosB + CosA sinB

= sin5pi/6cospi/4 + cos5pi/6sinpi/4

= (1/2)(sqrt2/2 + ( -sqrt3 /2)sqrt2/2

= sqrt(2)/4 - sqrt(6)/4

= (sqrt2 -sqrt6)/4

b) sin( -7pi/12) = -sin(7pi/12)

7pi/12 = 3pi/12 + 4pi/12

use rthe sum and difference identities.

Sin (a + b) = sin a cos b + cos a sin b

- Sin 7pi/12 = -sin (3pi/12 + 4pi/12)
sin (7pi/12) = sin (pi/4 + pi/3)

we know values ofsin pi/4, cos pi/4, sin pi/3, cos pi/3

-sin 7pi/12 = -sin (pi/4 + pi/3) = -sin pi/4*cos pi/3- cos pi/4*sin pi/3
... = -sqrt2/2*1/2 - sqrt 2/ 2 * sqrt 3 / 2
.... = -sqrt 2 / 4 - sqrt 6 / 4
... = -(sqrt 2 + sqrt 6) /4